BUFFER-STOCK SAVING AND THE
LIFE CYCLE/PERMANENT INCOME HYPOTHESIS*
Published in the Quarterly Journal of Economics
1997(1), Volume CXII, Pages 1-56
Christopher D. Carroll
Johns Hopkins University
ccarroll@jhu.edu
First Draft: July, 1990
This Draft: August 13, 1996
Abstract
This paper argues that the typical household’s saving is better described by a “bufferstock” version than by the traditional version of the Life Cycle/Permanent Income
Hypothesis (LC/PIH) model. Buffer-stock behavior emerges if consumers with
important income uncertainty are sufficiently impatient. In the traditional model,
consumption growth is determined solely by tastes; in contrast, buffer-stock consumers
set average consumption growth equal to average labor income growth, regardless of
tastes.
The model can explain three empirical puzzles: the “consumption/income
parallel” of Carroll and Summers [1991]; the “consumption/income divergence” first
documented in the 1930's; and the temporal stability of the household age/wealth profile
despite the unpredictability of idiosyncratic wealth changes.
Keywords: consumption, saving, precautionary saving, buffer-stock saving, uncertainty
JEL Codes: D91 E21
*Thanks to Olivier Blanchard, Sean Craig, Angus Deaton, Karen Dynan, Eric Engen, John Gruber, Miles Kimball,
James Poterba, Ranil Salgado, Lawrence Summers, Jonathan Skinner, David Weil, David Wilcox, Stephen Zeldes, and
the members of the MIT Money and Public Finance lunches for constructive comments. Remaining errors are my own.
-1-
I.
Introduction
Of the consumers who participated in the Federal Reserve Board's 1983 Survey of
Consumer Finances,
43 percent
said that being prepared for emergencies was the most
important reason for saving. Only 15 percent said that preparing for retirement was the most
important saving motive.1 These are not the answers that standard interpretations of the Life
Cycle/Permanent Income Hypothesis (LC/PIH) model of saving would lead one to expect.
This paper will argue, however, that such responses, and a wide range of other evidence,
are consistent with a version of the LC/PIH model in which consumers face important income
uncertainty, but are also both “prudent,” in Miles Kimball’s [1990b] sense that they have a
precautionary saving motive, and “impatient” in the sense that if future income were known with
certainty they would choose to consume more than their current income.2
Under these
conditions, consumers may engage in what I call “buffer-stock” saving behavior.3 Buffer-stock
savers have a target wealth-to-permanent-income ratio such that, if wealth is below the target, the
precautionary saving motive will dominate impatience and the consumer will save, while if
wealth is above the target, impatience will dominate prudence and the consumer will dissave.4
I first describe the main properties of the buffer-stock model in an infinite-horizon
context where labor income growth is constant. The model’s most surprising feature is its
implication that, even with a fixed aggregate interest rate, if consumers are sufficiently
“impatient,” average consumption growth will equal average labor income growth, either for
individual households or for aggregate consumption.5 This is true even though the consumers in
1
Summarized in Avery and Kennickell [1989]. The other possible categories were "To buy something or for the family" (29%),
and "Investment" (7%). Later SCF surveys produced similar results.
2
This desire to borrow can be due either to a high time preference rate or to high expected income growth.
3
Prudence, in Kimball’s sense of having a utility function with a positive third derivative, is not by itself sufficient to generate
buffer-stock saving. The utility function must also exhibit Decreasing Absolute Prudence, as does the Constant Relative Risk
Aversion (CRRA) utility function used in this paper. See Kimball [1990a, 1990b] for arguments that Decreasing Absolute
Prudence is a natural condition to require of utility functions.
4
The proof that a target wealth-to-income ratio exists, and is stable, is contained, along with many other derivations and proofs,
in a companion paper to this one, Carroll [1996]. For a copy of that paper, contact the author.
5
Standard general equilibrium models imply the economy will converge to a steady-state in which the growth rate of
consumption equals the growth rate of income. However, the mechanism by which this is achieved is through the dependence of
the interest rate on the capital stock; in the model considered here, this channel is short-circuited by assuming a fixed aggregate
-2-
the model behave according to the standard Euler equation which has been widely thought to
imply that consumption growth depends only on tastes, and not on the growth rate of income.
The problem in previous work has been in the common assumption that the second-order
variance term in the log-linearized version of the Euler equation can safely be ignored. In fact,
this variance term is an endogenous equilibrating variable: it will, on average, take on whatever
value is required to cause average consumption growth to equal average income growth.
I also present simulation evidence documenting other major differences between
implications of the buffer-stock model and what I will refer to as the “standard” model (either the
perfect-certainty model with Constant Relative Risk Aversion utility, or the Certainty Equivalent
(CEQ) model in which utility is quadratic). In comparison with the standard model, the bufferstock model predicts a much higher marginal propensity to consume out of transitory income, a
much higher effective discount rate for future labor income, and a positive rather than negative
sign for the correlation between saving and expected labor income growth.
Whether these results for the infinite-horizon model carry over to the finite-horizon
context cannot be determined analytically. The next section of the paper therefore solves a
finite-horizon version of the model under the same baseline parameter values used in the infinitehorizon model, but with age/income profiles roughly calibrated to U.S. household-level data. I
show that this configuration of the model generates buffer-stock saving behavior over most of
the working lifetime until roughly age 45 or 50, and behavior that resembles the “standard”
LC/PIH model only for roughly the period between age 50 and retirement.
I then argue that the finite-horizon version of the model can explain three major stylized
facts which, combined, cannot be explained by the principal alternative models: the “standard”
LC/PIH model, a Keynesian alternative to the standard LC/PIH framework, or the Campbell and
Mankiw [1989] combination of these two models.
First of the three stylized facts is the
“consumption/income parallel” documented by Carroll and Summers [1991] and Carroll [1994]:
when consumption is aggregated by groups or by whole economies it closely parallels growth in
income over periods of more than a few years. The consumption/income parallel, inconsistent
interest rate.
-3-
with the standard LC/PIH framework, is explained in the buffer-stock model as the result of
consumers’ impatience and their prudent unwillingness to borrow.
The second fact is the
“consumption/income divergence” that emerges from microeconomic consumer surveys: For
individual households, consumption is often far from current income, implying that the aggregate
consumption/income parallel does not arise from high frequency tracking of consumption to
income at the household level.
The consumption/income divergence, inconsistent with the
Keynesian model, is explained with essentially the same logic Friedman used long ago:
consumption does not respond one-for-one to transitory shocks to income because assets are
used to buffer consumption against such shocks.
The final set of stylized facts is about the patterns of wealth accumulation over the
lifetime. I show that the standard LC/PIH model implies that the productivity growth slowdown
after 1973 should have resulted in massive increases in household wealth-income ratios (because
slower expected growth implies lower current consumption). I then demonstrate that the modest
observed changes in the actual age/median-wealth profile match the predictions of the bufferstock model. Finally, I argue that the extraordinarily high volatility [Avery and Kennickell,
1989] of household liquid wealth is difficult to explain with either the LC/PIH or the Keynesian
models, but is a natural implication of a model in which the principal purpose of holding wealth
is so that it can be used to absorb random shocks to income.
Although the implications of the buffer-stock version of the LC/PIH model differ in
important respects from standard modern versions of the LC/PIH model, careful reading of
Friedman [1957] suggests that the buffer-stock version of the model represents a close
approximation to his original ideas.
Direct quotations from Friedman will illustrate the
similarities between his views and the implications of the buffer-stock model, and some of the
empirical discussion will parallel arguments that Friedman used long ago to justify his
conception of the Permanent Income Hypothesis.
The emphasis on Friedman is not meant to suggest that there has been no progress since
his book.
The buffer-stock model presented here owes much to the insights of Kimball
[1990a,b], Zeldes [1989a], and Deaton [1991], who have all emphasized the importance of
precautionary motives for saving. Indeed, the model presented here is structurally similar to the
-4-
model of Zeldes [1989a], with the important difference that I assume impatient consumers.
Alternatively, the model is similar to Deaton’s [1991] except that I do not directly impose
liquidity constraints and my model has independent transitory and permanent shocks.
The rest of the paper is organized as follows. Section II sets forth the basic intertemporal
optimization model, briefly describes the method of solution, discusses parametric assumptions,
and, for reference purposes, presents what I will call the “standard” LC/PIH model. Section III
describes the solution to the infinite-horizon version of the model. Section IV presents the finitelife solution to the model under age/income profiles calibrated from US data and argues that the
resulting behavior fits the stylized facts about household consumption, saving, and wealth better
than the alternative models. Section V contains a brief discussion the recent literature on models
related to the buffer-stock model, and Section VI concludes.
II.
The Basic Model
II. A. Solving The Model
I assume that the consumer solves the intertemporal optimization problem:
T
(1)
max Et
∑
i-t u(C )
i
i=t
such that
Wt+1
= R[Wt + Yt - Ct]
Yt
= Pt Vt
Pt
= Gt Pt-1 Nt,
where Y is current labor income; P is “permanent labor income” defined as the labor income
that would be received if the white noise multiplicative transitory shock to income, V, were
equal to its mean value of one; N is a lognormally distributed white noise mean one
multiplicative shock to permanent income; G = (1+g) is the growth factor for permanent labor
income; W is the stock of physical net wealth; R = (1+r) is the (constant) gross interest rate; and
= 1/(1+ ) is the discount factor where is the discount rate.
Optimal consumption in any period will depend on total current resources (or “gross
-5-
wealth”), the sum of current assets and current income, which, following Deaton [1991], I will
call X:
(2)
Xt = Wt + Yt.
The evolution of gross wealth is given by:
(3)
Xt+1 = R[Xt - Ct] + Yt+1.
Carroll [1996] demonstrates that this problem can be rewritten by dividing through all variables
by the level of permanent labor income. Defining lower case variables as the upper case variable
divided by the current level of permanent income (i.e. ct = Ct/Pt), the general Euler equation for
consumption then becomes:
(4)
1
= R Et-1 [{ ct[R [xt-1 - ct-1] / G Nt + Vt] GNt / ct-1 }
-
]
In the last period of life, it is optimal to consume everything, cT[xT] = xT. Thereafter,
recursion on equation (4) implicitly defines a rule for the consumption ratio as a function of the
gross wealth ratio in each period back to the beginning of life. Unfortunately, under general
forms of income uncertainty the consumption rules do not have analytical formulas, so they must
be approximated by numerical methods. The details of the method of numerical solution are
contained in Appendix I.
II. B. Parameter Values
I will solve the model and present most of my results for a single baseline set of
parameter values, but will also present a summary of results for alternative choices of all
parameter values so that readers who differ with any particular parametric choice can determine
how sensitive my results are to changes in that parameter. The baseline values for characterizing
the distribution of income shocks will be the same as in Carroll [1992]; using data from the
Panel Study of Income Dynamics, that paper found that household income uncertainly was well
-6-
captured by a process that took the form:
V~
0
Z
with probability p
with probability (1-p) ,
ln Z ~ TN(
2
ln N ~ TN(
2
ln Z/2,
ln N/2,
2
ln Z),
2
ln N),
where TN signifies a truncated normal distribution (in practice, I truncate at three standard
deviations above and below the mean), and
ln Z
is chosen to make Et Vt+1 = 1. The choice of
mean for ln N was similarly motivated by the wish to make Et Nt+1 = 1 regardless of the choice
of
2
ln N;
this simplifies the analysis of the effect of changes in
2
ln N
on wealth. The probability
of zero income was estimated at about p=0.5 percent per year, and the standard deviation of both
transitory and permanent income shocks were estimated to be around 0.1 percent per year, after
crudely accounting for the effects of measurement error.
The expected growth rate of labor income appropriate for calibrating this model is the
growth rate in household labor income for working households. Carroll and Summers [1991]
provide evidence that, over long periods, household income growth can be characterized as the
sum of aggregate productivity growth and a household-specific component that reflects such
factors as increasing job tenure, seniority, and experience. If we assume very conservatively that
each of these factors contributes 1 percent annually to household income growth, the appropriate
baseline assumption for the growth rate of household labor income is 2 percent per year; this will
be the baseline assumption for the infinite-horizon version of the model. For the finite-horizon
version of the model used later, the pattern of income growth over the lifetime will be calibrated
explicitly using household data.
Carroll [1992] made the strong assumption that the time preference rate was 10 percent
per year; this is a substantial departure from common assumptions in the economic literature.
Much macroeconomic research assumes a discount rate of one percent per quarter, or about 4
-7-
percent per year.6 In order to emphasize that the results obtained in this paper are not the result
of extreme assumptions about the time preference rate, the baseline value of the discount rate
assumed in this paper will be 4 percent per year.7
The baseline interest rate will be zero, also following Carroll [1992]. The asset in this
model is perfectly riskless and perfectly liquid; the closest proxy is probably the three-month Tbill, whose after-tax rate of return over the postwar period has been roughly zero. Results for
interest rates of two percent and four percent are qualitatively similar, and will be presented
later.8
Estimates of the coefficient of relative risk aversion vary widely. Empirical estimates
above 6 have often been obtained (see, e.g., Mankiw and Zeldes [1991]),9 but many economists
believe that values above about 5 imply greater risk aversion than is plausible. At the other
extreme, log utility, which is the limit of the CRRA utility function as
approaches one, is a
common assumption because under some circumstances it is analytically tractable. The baseline
value of
for this paper will be
= 2, toward the low end of the usual range in order to avoid
exaggerating the magnitude of precautionary saving effects.
II. C. Comparison to Previous Work
This model is similar in many respects to models considered by Deaton [1991] and
Zeldes [1989a].
The finite-horizon version differs from Zeldes’s model primarily in the
parametric assumptions about income uncertainty and tastes. Zeldes did not calibrate his model
explicitly using panel data on household income from the PSID, and, more important, assumed
that consumers were substantially more patient than I assume here; Zeldes also did not examine
6
Examples include Kydland nad Prescott [1982], Hansen [1985], and Benhabib, Rogerson, and Wright [1992].
7
I do not appeal here to empirical evidence on the discount rate because I will argue below that one of the implications of the
theoretical results in this paper is that the empirical methods that have been used to estimate discount rates, as in, e.g., Lawrance
[1991], are fundamentally flawed.
8
Readers accustomed to general equilibrium representative agent models may object to having such a large gap between the
interest rate and the rate of time preference. My view is that this model is the right description of the behavior of the typical
consumer, but probably not the right model for understanding where most of the aggregate capital stock comes from. See the
conclusion for a more extended discussion of this point.
9
However, see below for a critique of the method of estimating
in this and other similar papers.
-8-
an infinite-horizon version of his model. The infinite-horizon version of my model differs from
Deaton’s [1991] in simultaneously incorporating both transitory and permanent shocks to
income; because of the presence of zero-income events; and because Deaton imposes explicit
liquidity constraints. However, as Zeldes [1989a], and, earlier, Schechtman [1976] have pointed
out, the combination of the assumption that income can go to zero in each period with the
assumption that consumption must remain strictly positive is sufficient to guarantee that
consumers will never borrow.10
Despite the formal modelling differences, I view the infinite-horizon version of this
model and Deaton’s as close substitutes because very similar household behavior emerges from
the two models. One insight this similarity provides is that Deaton’s qualitative results are
attributable mainly to his assumption that consumers are impatient rather than to the assumption
of liquidity constraints. An analytical convenience of this formulation over Deaton’s is that,
here, consumption always obeys the standard Euler equation linking the marginal utility of
consumption in one period to marginal utility in adjacent periods. In Deaton’s model, the usual
Euler equation is violated whenever the liquidity constraints are binding.
For purposes of comparison, a brief description is in order of the model I will refer to as
the “standard” model. The specific model I will consider is the perfect certainty version of the
CRRA model described above, i.e. the model that would apply if the transitory and permanent
shocks were known in advance to always be equal to their expected values, Vt = Nt = 1 for all t,
although results in most cases would be very similar for the version of the model with quadratic
utility and uncertainty. Defining human wealth H as the present discounted value of the expected
stream of future income, and denoting the marginal propensity to consume by k, the solution to
this model in the finite and the infinite horizons is given by:
10
They refuse to borrow essentially for precautionary reasons, fearing the consequences of borrowing and then earning zero
income indefinitely, so that eventually consumption is driven to zero. The no-borrowing result is less special than it may appear,
however; the qualitative characteristics of the model are unchanged if the lower bound on income is positive. In that case,
consumers will sometimes borrow, but will never borrow more than the present discounted value of the minimum possible future
income stream. In effect, this amounts only to a shift in the horizontal axis for the problem. For a more detailed discussion, see
Carroll [1992].
-9Finite Horizon
Ct
Infinite Horizon
= kt[Xt + Ht]
Ct
Yt+1
= G Yt
Ht
=
•
T
Ht
=
Â
i =t +1
Ri-tYi
=
(1-[R-1( R)]1/ )
______________
k
Â
i =t +1
≈
kt
= k[Xt + Ht]
Ri-tYi
Yt
(r - g)
= (1 - [R-1( R)1/ ])
(1-[R-1( R)1/ ]T-t+1)
III.
Characteristics of the Solution
III. A. The Optimal Consumption Rule and the Consumption Euler Equation
In the version of his model with only permanent shocks, Deaton shows that if the change
in permanent income is distributed lognormally with variance
2
ln N,
then, making the usual
approximations that ln[R]≈ r, ln[ ] ≈ - , and ln[G] ≈ g, the successive consumption rules ct[xt],
ct-1[xt-1], ... converge if11
(5)
-1(r
- ) + ( /2)
2
ln N
<g-
2
ln N
/ 2.
Carroll [1996] proves that this same condition guarantees convergence of the consumption rules
in the model in this paper.12
This formula differs slightly from Deaton’s, which lacks the - 2ln N /2 term on the right hand side. The difference is merely
notational: Deaton calls the mean of his lognormal permaennt income shock g, while in my framework the mean of log(GN) is
g- 2ln N / 2. My definition was chosen because it implies that the expected value of the permanent shock is 1 regardless of the
11
assumption about the variance of the permanent shocks.
12
The exact condition (without approximations) is (R )Et[GN t+1]- < 1.
- 10 -
The intuition for this equation is easiest to grasp if we assume
the standard model, the growth rate of consumption is
with zero assets.
-1(r
2
ln N
= 0 momentarily. In
- ).13 Now consider a consumer
Because the PDV of income must equal the PDV of consumption, if
consumption growth will be slower than income growth over the remainder of the lifetime (i.e. if
(5) holds), the level of consumption today must be higher than the level of income today. Thus,
the condition boils down to whether the consumer is sufficiently impatient that he would wish to
dissave (or borrow) today to finance current consumption, if future income were perfectly
certain. The more general case, with the
2
ln N/2
terms reflects, on the left hand side of equation
(5), the additional consumption growth induced by the permanent income shocks, and, on the
right hand side of the equation, the reduction in the mean growth of the log of income necessary
to maintain EtNt+1 = 1 (without this adjustment EtNt+1 would increase with
2
ln N).
Equation (5)
is the condition referred to informally in the introduction as the “impatience” assumption,
although note that this equation can be satisfied by consumers who do not discount future utility
at all ( = 0) but who face positive income growth.
Many of the important results from the buffer-stock model can be understood by
considering the log-linearized consumption Euler equation, which takes the form
(6)
Et ln Ct+1 ≈
-1(r
- ) + ( / 2) vart ( ln Ct+1) + et+1
if shocks to consumption are lognormally distributed.14
The bulk of previous work on
consumption (see, e.g., Hansen and Singleton [1983], Hall [1988], Zeldes [1989b], Lawrance
[1991]) has essentially ignored the expected variance term in the consumption Euler equation,
assuming it to be either a constant or zero.
13
Again, this is an approximation. The exact result is that (ct+1/ct)=(R )-1/ . Henceforth, in the text and in figures, I will
approximate log (R ) with (r- ) and log G with g without further comment, although in all the calculations the correct (not
approximate) formulae are used.
14
See, e.g., Deaton [1992]. If shocks to consumption are not lognormally distributed, a similar equation can be derived using a
Taylor expansion of the Euler equation, see Dynan [1993]. The formula in equation (6) was used because it is more intuitive than
the expression from the Taylor expansion.
- 11 -
Figure Ia summarizes many of the important features of the buffer-stock model. The
curve labelled “ (x) = Et
ln Ct+1” corresponds to the expectation of consumption growth
(calculated numerically using the converged consumption rule) as a function of the consumer’s
gross wealth-to-income ratio. The horizontal line drawn at
1(r
- ) indicates the growth rate of
consumption that would prevail in a standard model with CRRA utility and baseline parameter
values but with no labor income uncertainty.15 The other horizontal line is the expected growth
rate of permanent income, g’ = Et
greater than
1(r
- )-
2
ln N/2,
ln Pt+1 = g -
2
ln N
/2, which under these parameter values is
thus guaranteeing that these consumers are “impatient” in the
required sense of equation (5). The vertical line labelled “x*” represents the target value of the
gross wealth ratio, i.e. x* is the xt such that Et xt+1 = xt.
The first point the figure illustrates is the inadequacy of the common assumption that the
variance of consumption growth is constant or zero: The gap between the Et ln Ct+1 curve and
the
1(r
- ) line is strongly declining in the level of the gross wealth ratio.16 This happens for
the intuitive reason that consumers with less wealth have less ability to buffer their consumption
against shocks to income. More formally, the declining variance is a result of the fact that the
optimal consumption rule is strictly concave; in other words, the marginal propensity to consume
is a strictly decreasing function of the level of wealth. (Carroll and Kimball [1996] prove the
strict concavity of the consumption function for a wide class of problems which includes this
one). The link between concavity and the variance term stems from the fact that at low levels of
wealth, the marginal propensity to consume is high, so for a poor consumer a given amount of
variation in income will induce a larger amount of variation in consumption than the same
income variation would induce for a consumer with more wealth and thus a lower MPC.
Carroll [1996] formally proves a variety of propositions about this figure. First, as xt
0,
15
In fact, the CRRA model with certain income does not have a well-defined solution for the baseline income growth and interest
rate parameter values, because with an interest rate less than the income growth rate, the present discounted value of future
income is unbounded. However, for a finite-horizon version of the certainty model where the horizon is arbitrarily long, it
remains true that the growth rate of consumption will be given by 1(r - ).
Strictly speaking, the gap between (x) and 1(r- ) is not exactly proportional to the variance of expected consumption
growth, because equation (6) is an approximation. Qualitative statements about the gap and the variance are interchangable,
however, so as a heuristic tool I will speak of the gap interchangably with the variance.
16
- 12 -
the expected rate of consumption growth goes to infinity (although for graphing purposes the
expected consumption growth locus is truncated at 10 percent). This is essentially because as xt
0, Ct
0 and therefore log Ct -∞. Second, as xt
approaches
1(r
∞, the expected growth rate of consumption
- ), the growth rate that prevails in the perfect certainty model. This is because
as wealth approaches infinity, the proportion of future consumption the consumer expects to
finance out of his (uncertain) labor income becomes infinitesimal, so for all practical purposes
labor income uncertainty becomes irrelevant.17 Third, there exists a target wealth-to-income
ratio x* such that if xt = x*, Etxt+1 = x*, and that target is “stable” in the sense that if xt > x*,
Etxt+1 < xt and vice versa.
(This result justifies the directional arrows on the expected
consumption growth locus.)
The fourth proposition is a correction of a proposition in Carroll [1992]. That paper
argued that at the target gross wealth ratio x*, expected consumption growth was
“approximately” equal to expected permanent income growth,
Et [ ln Ct+1 | xt = x*] ≈ Et
ln Pt+1.
Carroll [1996] shows that a more appropriate approximation is
(7)
Et [ ln Ct+1 | xt = x*] ≈ Et
ln Pt+1 +
’’[x*]
Et
(x t+1 - x*) 2
2
where [x] = log c[x].18 Carroll [1996] uses the proof in Carroll and Kimball [1996] of the strict
concavity of the consumption function to show that the ’’[x*] term is strictly negative. Thus,
expected consumption growth for consumers holding x* is strictly less than the expected growth
rate of permanent labor income.
This is visible in the figure from the gap between the
17
This proof requires an additional restriction on parameter values, g < r, which is not satisfied by my baseline parameter values
but is satisfied by some of the alternative parameter values considered later.
18
The intuition here is Jensen’s inequality: because the expected growth curve is convex, the average growth of consumption for
consumers distributed around the target wealth will be greater than the expected growth rate at the target.
- 13 -
intersection of the x* line with the Et [ ln Ct+1] locus and the intersection of the x* line with the
Et [ ln Pt+1] line.
Figure Ib provides an example of how to perform experiments with the figure. The solid
lines represent a blowup of the middle portion of figure Ia, while the dashing lines show how the
figure changes if g, the expected growth rate of labor income, declines from the baseline value of
g=.02 to a new value of g=.005 annually. (The dashing horizontal line indicates the new, lower
expected income growth rate, g2’ = g2 -
2
ln N
/2 = .005 - .005 = 0). Even though the expected
income growth rate does not appear directly in the Euler equation for consumption growth (6),
the locus depicting the expected growth rate of consumption does shift, from its original position
1
(x) to
2
(x), the downward-sloping dashing curve. The expected consumption growth curve
shifts because the expected variance term in (6) at a given x changes; the new optimal
consumption rule will differ from the old one, so at a given x the same amount of variation in
income can produce a different amount of variation in consumption. The new target wealth ratio
x*2 is greater than the original one; thus, as one would suspect, when consumers expect slower
income growth, they hold more wealth. This is the manifestation of the “human wealth effect” in
this model.
The new
2
(x) locus in Figure Ib was constructed by solving the whole model under the
new growth rate assumption and again numerically calculating the expectation of consumption
growth as a function of x. It would be convenient if there were a shortcut to this laborious
process, but unfortunately there appears to be no other way to obtain accurate quantitative
answers to questions about how target wealth changes when parameter values change. On the
other hand, there is a simple procedure which appears always to give correct qualitative answers
to such questions. Define [x*] = [Et ln Ct+1 | x = x*] - Et ln Pt+1; that is,
corresponds to the
last term in equation (7), the gap between expected consumption growth and expected income
growth for a consumer with wealth equal to target wealth. Now, to determine how a given
parameter affects target wealth, shift only the curves which directly reflect the parameter in
question, and find the value of x* which leaves [x] the same as under the original parameter
values. Figure Ic illustrates how this procedure would apply to the experiment that is performed
“correctly” in figure Ib, a decline in the expected growth rate of income. Because the only locus
- 14 -
that directly reflects the growth rate is the growth curve itself, that is the only curve in the figure
that needs to be shifted. The new x*, x*’2, is drawn at the point that leaves the gap
[ ln Ct+1 | x = x*] and Et
between Et
ln Pt+1 unchanged. As with the “correct” procedure in Figure Ib, the
qualitative answer this exercise yields is that a decline in the growth rate of income produces an
increase in the target wealth ratio.
III. B. The Steady-State Distribution of Assets
The description of the implications of the model thus far has focused on features and
implications of the optimal consumption rule. This is in keeping with most previous research,
including Zeldes [1989a] and Kimball [1990a], who, for instance, examine the effect of
uncertainty on the marginal propensity to consume at given levels of wealth or consumption, but
do not examine what levels of wealth and consumption will prevail. As a result, the main
implications they are able to draw about the differences between their models and standard
models are qualitative - for their models, the marginal propensity to consume out of transitory
income will be higher, consumers will hold more wealth, the expected growth rate of
consumption will be greater, than for the standard model. However, to compare the model to
quantitative empirical results, it is necessary to be able to answer the question how much greater
will the MPC be, on average? How much higher is wealth, typically? How much faster is
consumption growth, on average? And how do the answers to these questions depend on
assumptions about the degree of income uncertainty, the value of taste parameters, and the
expected growth rate of income?
To answer such questions it is necessary to compute the distribution of wealth implied by
the model. Presumably one reason Zeldes did not attempt this is that in a finite-horizon version
of the model, the consumption rules and the distribution of wealth change in every period of life,
so the answers to those questions would have been different for every period of life. It would be
difficult, therefore, to summarize the results.
Things are potentially much simpler in the infinite-horizon context. Clarida [1987]
showed that in a similar model with liquidity constraints, no permanent shocks, and no growth,
the distribution of assets will be ergodic, converging toward a fixed “steady-state” distribution.
If the permanent shocks in this model are removed (i.e. Ni,t=1 for all t), this model satisfies the
- 15 -
critical condition for ergodicity posed in Clarida [1987] (see Carroll [1996] for a proof).
Unfortunately, I have been unable to construct a general proof of ergodicity for the model with
permanent shocks. Nonetheless, for any particular converged consumption rule, if an ergodic
distribution exists, it can be found by numerical methods. (See Appendix II for a description of
the numerical procedure).
Figure II shows the evolution of the distribution of the net wealth ratio over time in a
simulated economy containing 20,000 households who all behave in every period according to
the converged consumption rule derived under the baseline parameter values. Consumers in the
simulation begin life with zero assets and permanent labor income in the first period of Pi,1 = 1
for every household i. In each subsequent period of life, each household receives independent
shocks drawn from the income distributions described above.19 What is depicted in Figure II is
the temporal evolution of the across-household distribution of the net wealth ratio, where net
wealth is defined as the remaining wealth in a period after the household has drawn all shocks
and has consumed the desired amount. (This concept corresponds better to the data typically
reported in wealth surveys than does the gross wealth ratio x used heretofore in this paper.)
Convergence of the distribution toward the steady state distribution is rapid. From a
starting value of zero, after only three periods of life the mean net wealth ratio is 0.27, compared
to an estimated steady-state value of 0.34. The speed of convergence to the steady-state is
interesting because if convergence is rapid, it is more likely that any steady-state results derived
will also apply, at least roughly, to collections of consumers out of steady-state.
III. C. The Relationship Between Consumption Growth and Income Growth
Table I presents some summary statistics about the “steady-state” behavior of collections
of buffer-stock consumers who all have identical preferences. Each row presents results when
all parameters are at their baseline values except the parameter designated in the first column,
which takes the indicated value. The most striking result is shown in Columns 2, 3, and 4: the
expected growth rate of households’ consumption always matches the expected growth rate of
their permanent labor income, and the growth rate of their aggregate consumption always
19
I assume that there are no aggregate shocks.
- 16 -
matches the growth rate of their aggregate labor income. This section explains how these results
arise.
Consider a collection of ex ante identical buffer-stock consumers indexed by i who as of
period t have achieved among them the steady-state distribution for ci,t, the ratio of consumption
to permanent income. Designating E.,t as the expectation taken across all households as of time
t, the average expected growth rate of consumption for these consumers is given by:
(8)
E.,t
ln Ci,t+1
= E.,t [ ln Ci,t+1 - ln Ci,t]
= E.,t [ ln ci,t+1GNi,t+1Pi,t- ln ci,tPi,t]
= E.,t [ ln GNi,t+1 + ln ci,t+1 - ln ci,t]
= E.,t [ ln GNi,t+1] + E.,t [ln ci,t+1] - E.,t [ln ci,t]
= E.,t [ ln GNi,t+1] = g -
2
ln N/2,
where the last equality follows because in steady-state the average value of the consumption ratio
does not change from one period to the next, E.,t [ln ci,t+1] = E.,t [ln ci,t].
In the same notation, equation (6) can be rewritten as:
(6’)
E.,t
ln Ci,t+1 ≈
-1(r
- ) + ( /2) E.,t var ( ln Ci,t+1).
We now have two expressions, equations (8) and (6’), for the expected growth rate of
consumption, taking expectations across consumers at a point in time. The two equations share a
single endogenous variable, the expected variance of consumption growth.
Obviously this
means we can solve for the endogenous value of the expected variance:
(9)
E.,t [ var i,t( ln Ci,t+1) ] ≈ (2/ ) [g -
2
ln N/2
-
1(r
- ) ].
Of course, as Figure I vividly illustrates, the variance of consumption growth is also a
negative function of the level of wealth. Using that fact in conjunction with equation (9) yields
intuitive predictions. For example, at higher interest rates the right hand side of equation (9) will
- 17 -
be smaller, corresponding to a smaller average consumption variance and thus a higher average
level of wealth; in other words, the interest elasticity of average wealth is positive.20 Similar
logic can be used to show that wealth is higher for consumers who discount the future less (
falls) or for consumers with lower expected permanent income growth (the human wealth effect).
The coefficient of relative risk aversion has offsetting effects: A higher
represents a stronger
precautionary saving motive (reflected in the 2/ term), and on its own would increase average
wealth. But a higher
(reflected in the
1
also corresponds to a lower intertemporal elasticity of substitution
(r - ) term), which should result in lower wealth if consumers are impatient.
The overall effect of on wealth is therefore ambiguous.
Because the average growth rate is not the same as the growth rate of the average, the
above proof that E.,t
ln Ci,t+1 = E.,t
ln Pi,t+1 does not prove that
ln E.,t Ci,t+1 =
ln E.,t Pi,t+1.
Yet Table I showed that in the simulations both propositions held true. It is surprisingly difficult
to prove that the average growth rate of aggregate consumption is equal to the average growth
rate of aggregate income in the buffer-stock model. Because the proof is difficult but not
enlightening it is not presented here; interested readers can find it in Carroll [1996].
A word is in order about the subtle but important difference between the kind of analysis
contained in the previous section and embodied in Figure I, and the kind of analysis just
presented that can be done using equations (6’) and (9). Figure I and its discussion reflected only
statements about the optimal behavior of an individual consumer; nowhere were implications
about aggregates or averages across consumers derived or discussed. On the other hand, the
logic used to obtain equation (9) relied critically on an assumption that there was a population of
consumers across whom the distribution of consumption (and other variables) had reached its
ergodic steady-state.
20
The reasoning relating parameter values to mean wealth in this sentence and the rest of the paragraph cannot be formally
justified, for several reasons. For example, the demonstration in Figure I that vart ( ln Ct+1) is a negative function of xt does
not absolutely guarantee that when the equilibrium variance falls mean wealth must rise. That would only follow with absolute
necessity if the new steady-state distribution of x were identical to the original distribution except for a location parameter, and if
the relationship between vart ( ln Ct+1 ) and xt were linear, neither of which is true. Furthermore, even equation (9) itself relies
on approximations. Equation (9), and the methods for reasoning about average wealth from it, should be viewed as a heuristic
tool rather than a rigorous analytical framework. That said, I have found no parameter values for which this kind of reasoning
from equation (9) gives the wrong answer.
- 18 -
III. D. Implications for Empirical Research
The most important implication of equation (9) is that typical methods of Euler equation
estimation, on either household or aggregate data, yield meaningless results if the consumers
involved are buffer-stock savers, because typical methods assume that the variance term in the
Euler equation is either zero or a constant. This subsection illuminates the potential pitfalls for
Euler equation estimation using examples, first from the literature on Euler equation estimation
with household data and then from the literature on aggregate Euler equation estimation.
Lawrance [1991] estimates consumption Euler equations across households of different
educational levels, assuming a constant value of the variance term across households.
Simplifying considerably, she finds that consumption growth is faster for households with
greater education, and concludes that education must be correlated with the pure rate of time
preference, . However, a large literature in labor economics has established that households
with greater education have faster labor income growth. The dependence of consumption growth
on income growth in the bufer-stock model strongly suggests that Lawrance’s estimates of
might simply be proxying for the effects on consumption growth of predictable differences in
income growth.
Because the issues here are both subtle and important, it is vital to be perfectly clear
about the nature of the problem. A simple example will illustrate how results like Lawrence’s
could arise in a buffer-stock framework despite identical time preference rates across
households. Suppose that the population consists of two kinds of consumers, H and L, identical
in every respect (including taste parameters) except that the mean growth rate of permanent labor
income is higher for consumers in group H than for consumers in group L, gH > gL. Suppose
further that the two groups of consumers have respectively converged to their steady-state wealth
distributions. Now imagine estimating an equation of the form:
ln Ci,t+1 =
0
+
1
Ei,t ri,t+1 +
1
Di +
i,t+1
across this whole population of consumers, with the dummy variable D equal to one for
consumers of type H and 0 for consumers of type L. The estimated coefficient on Dii will be (gH
- 19 -
- gL), as can be seen by plugging equation (9) with different growth rates into equation (6’). The
Lawrence interpretation of this finding would be that consumers in group H have a lower
discount rate than consumers in group L, even though by assumption the data were generated by
consumers with identical time preference rates.
The problem is in the omission from the
estimating equation of the endogenous variance term, which buffer-stock theory indicates ought
to be correlated with Dii.. In econometric terms, this is an omitted variable problem, where the
omitted variable is correlated with the included variable D.
Even papers which explicitly acknowledge the presence and potential non-constancy of
the variance term face substantial problems. One of the earliest and best of these papers is
Dynan [1993]. She uses data from the Bureau of Labor Statistics’ Consumer Expenditure
Surveys to calculate consumption growth rates and variances of consumption growth rates for
different groups of households and estimates an equation of the form
ln Ci,t+1 =
0
+
1
Ei,t ri,t+1 +
2
E.,t var( ln Ci,t+1) +
i,t+1
by instrumental variables, using as instruments the head of household’s education, occupation,
age, and a variety of other characteristics. She finds a coefficient close to zero on both the Ei,t
ri,t+1 term and the E.,t var(
ln Ci,t+1) term.21 Dynan considers herself to be estimating equation
(6) and thus expects the coefficient on r.,t+1 to equal
to be ( /2). A coefficient estimate of zero for
risk aversion of
1
-1
and the coefficient on E.,t var(
ln Ci,t+1)
would therefore imply a coefficient of relative
= ∞, while a coefficient of zero on the variance term implies
= 0. However,
Dynan focuses on the coefficient estimate for E.,t var( ln Ci,t+1), and concludes that her data
provide no evidence for the existence of a precautionary saving motive (a precautionary motive
requires a strictly positive ).
If Dynan’s consumers were engaged in buffer-stock saving, however, coefficient
21
An important subtlety: for simplicity, I will assume that Dynan’s instruments are effectively isolating separate groups of
consumers with different group characteristics. In this case the E.,t notation represents the instrumented value of the variable
whose expectation is being taken.
- 20 -
estimates of zero on both these terms would be unsurprising. The simplest example of how such
a result could arise is as follows. Consider a sample which contains households that fall into
several groups identifiable to the econometrician via instruments like Dynan’s (education groups,
say). Suppose that the households in the groups are identical in every respect except for the
interest rates they face and their rates of time preference. In the buffer-stock framework, on
average the impatient groups of consumers will end up holding less wealth. Their smaller
buffer-stocks reduce their ability to shield consumption against income shocks, resulting in a
r
larger value for 2 E.,t var( ln Ci,t+1). Under these circumstances, group membership would be
highly statistically significant (as Dynan’s instruments are) in a first-stage regression of the
variance term and the interest rate term on group dummies, yet all consumers would end up with
the same growth rate of consumption (their shared growth rate of income g) despite having
predictably different values of both E.,t var ( ln Ci,t+1) and E.,t ri,t+1. The regression’s intercept
term
0
would be estimated to equal g and the coefficients on the interest rate and variance terms
would both be estimated at zero, as Dynan found. Similar logic holds if the coefficient of
relative risk aversion varies across groups; the coefficient estimates for 1 and 2 would again be
zero. Taken as a whole, therefore, Dynan’s findings are actually supportive of a buffer-stock
model of precautionary saving, but only because, in contrast to the standard model, the bufferstock model is capable of explaining how both of her coefficient estimates could be estimated at
zero even if consumers in fact have a strong precautionary saving motive.
In principle, it is possible to estimate consumption Euler equations consistently across
groups of buffer-stock consumers, but the conditions required are quite stringent. In particular, it
is essential to have important and predictable differences across groups of consumers in both
interest rates and income growth rates, and no differences at all in tastes across groups. Without
variation in income growth rates, equation (9) indicates that the variance term and the interest
rate term should be perfectly collinear, preventing accurate coefficient estimation on either term.
Without variation in interest rates across groups, it is obviously not possible to estimate a
coefficient on the interest rate term.
The implications of this discussion for the estimation of consumption Euler equations
across households are thus rather grim, at least for groups of consumers who satisfy the
impatience condition (5). One might hope that the traditional Euler equation estimation methods
- 21 -
would at least be applicable to patient consumers who do not satisfy (5).
Unfortunately,
however, even for finite-horizon patient consumers the variance term in the Euler equation will
be a declining function of wealth, because the proof of the concavity of the consumption function
in Carroll and Kimball [1996] holds regardless of taste parameters. Because the amount of
wealth accumulation accomplished by a given age, and therefore the variance term in the Euler
equation, depends on the time preference rate, the time preference rate cannot itself be estimated
consistently using Euler equation methods across such groups of consumers. Similar logic
applies to the coefficient of relative risk aversion. Of course, as the level of the gross wealth
ratio gets very large, the variance of consumption growth approaches zero and these problems
disappear. In fact, the only consumers for whom it is absolutely clear that taste parameters can
be consistently estimated via traditional consumption Euler equation methods are consumers
who possess effectively infinite wealth.
The message of the buffer-stock model for the estimation of household taste parameters
is not entirely nihilistic, however. Equation (9) itself can be estimated, most easily by regressing
group variances of consumption growth on group income growth rates and group-specific
interest rates: this yields a direct estimate of (2/ ) and therefore of itself, assuming a that is
constant across groups.
One advantage of estimating (9) over direct estimation of the
consumption Euler equation is that consistent estimation of (9) need not require identical time
preference rates across groups. Indeed, if good instruments can be found for the growth rate of
income and for the interest rate, it should even be possible to use nonlinear methods to estimate
group time preference rates from equation (9).
A second, cruder test of the foundations of the buffer-stock model, and indeed of the
more fundamental property of the concavity of the consumption function, is suggested by Figure
I: simply examine whether the variance of consumption growth is higher for consumers with
lower wealth-to-permanent-income ratios. Perhaps the best empirical test along these lines
would be to look for consumers who experienced a major recent drop in their wealth (possibly
owing to a spell of unemployment), and to calculate the effects on the variance term and on
subsequent consumption growth.
Equation (9) and Figure I do not exhaust the empirical implications of the model for
household data. See Section V. for a brief discussion of a variety of recent empirical work which
- 22 -
matches the model to data in new ways that have little to do with Euler equation estimation.
Implications of the buffer-stock model for estimation of aggregate Euler equations are
similar to those for estimation across groups of households. If the population is normalized at 1,
the aggregate and average levels of consumption will be the same; designate both as E.,t+1 Ci,t =
C.,t. In that case the relationship between aggregate labor income growth g and aggregate
consumption growth in the “steady-state” can be written simply as:
(10)
ln C.,t+1 = g.
The mechanism by which the growth rate of aggregate consumption converges to the
growth rate of aggregate labor income is essentially the same as that which caused expected
household consumption growth within groups of buffer-stock consumers to converge to expected
household income growth:
Through the mediation of the endogenous variance term.
Households living in a more impatient country will have a lower value of household wealth, on
average, and therefore higher consumption variance, which will boost their consumption growth
enough to make consumption growth match income growth.
The model’s prediction that aggregate consumption growth should approximately equal
aggregate permanent labor income growth, at least in steady-state, may have the potential to help
explain many empirical failures of the standard Euler equation framework estimated using
aggregate data. The most obvious application is to the consumption/income parallel that Carroll
and Summers [1991] documented for cross-country aggregate data over periods of three to five
years and longer. Of course, standard growth models also predict that, in steady-state, income
growth and consumption growth will be “balanced.” The distinction between this model and the
standard model is that the transition to the steady-state is much faster here: Under baseline
parameter values, the transition half-life in the buffer-stock model is generally about two years;
in contrast, under standard parameter values the transition half-life in a Solow or CassKoopmans growth model is on the order of fifteen or twenty years, and some authors (Mankiw,
Romer, and Weil [1992] in particular) favor parameter values that generate a half-life that is even
longer. The slow rate of convergence in the standard model was a primary reason Carroll and
- 23 -
Summers [1991] rejected the idea that the consumption/income parallel evident over periods of 3
to 5 years was a reflection of balanced growth steady-states.
Another piece of evidence Carroll and Summers [1991] mustered against the CassKoopmans growth model as a description of aggregate consumption was that there is no apparent
relationship across countries between the average growth rate of aggregate consumption and
average country-specific interest rates.
This result is easily explained in a buffer-stock
framework as resulting from the endogeneity of the variance term, which is not observed in
aggregate data and is hence inevitably omitted from aggregate Euler equation estimation.22 As
equation (9) indicates, the omitted variance term is in theory negatively correlated with the
interest rate. In fact, equation (10) summarizes all the implications of the buffer-stock model for
the steady-state relationship between consumption growth and interest rates, tastes, income
uncertainty, and other variables: the coefficients on all such variables should be zero when the
consumption variance term is omitted from the estimating equation.
The model may also help to shed some light on the findings of Campbell and Mankiw
[1989], who estimated an equation of the form:
ln Ct+1 =
0
+
1
Et rt+1 + Et ln Yt+1 +
t+1
where Et ln Yt+1 was calculated using lagged variables that, according to traditional Euler
equation analysis, should be uncorrelated with current consumption growth. They found a
highly significant estimate of
in the vicinity of .5. Their interpretation was that half of
consumption is done by consumers who set consumption equal to income, while the rest is done
by consumers who obey the standard Euler equation (although they did not find robust evidence
of a positive coefficient on the interest rate as would be expected for the consumers who
putatively obey the standard Euler equation).
Suppose that in the postwar period aggregate labor income grew according to
22
It is important to recognize here that the right variable is the variance of consumption growth at the microeconomic level. The
omitted variance term therefore cannot be recovered from any kind of ARCH or GARCH estimation using aggregate data; it must
be calculated using household level data, if at all.
- 24 -
ln Yt = gt + et
where g represents the underlying rate of labor income growth and e represents transitory shocks
to income growth. Furthermore, suppose gt = g1 for t before 1973 and gt = g2 < g1 for t after
1973, reflecting the productivity growth slowdown in the post-1973 period. Now suppose that
some subset of Campbell and Mankiw’s instruments also experienced a regime shift in the post1973 period, and therefore instruments dated t will do a good job indicating whether the
economy is in the slow-growth or fast-growth regime; finally, suppose another subset of
instruments is highly correlated with et+1. The second-stage equation that Campbell and Mankiw
estimate is then in effect:
ln Ct+1 =
0
+
1
Et rt+1 + (gt+1 + e t+1) +
t+1
where gt+1 reflects the correlation of their instruments rates with underlying “permanent” growth
rate regime, and e t+1 reflects the correlation of their instruments with the predictable component
of transitory income growth. The buffer-stock model implies that, across steady-states, the
coefficient on gt+1 should be one. It is less clear what the model would imply about the
coefficient on e, although the MPC out of transitory income should be an upper bound on the
coefficient on e. Although it is not clear what the coefficient on g would be during the transition
period between the pre-1973 and post-1973 growth regimes, the relatively rapid convergence of
the buffer-stock model under baseline parameter values suggests that this transition period would
not last long, so the estimated coefficient on g should be close to one if separate coefficients
t+1
on g and e were estimated. However, when the two terms are combined into a single term for
predictable income growth, as Campbell and Mankiw do, any coefficient estimate between one
and the coefficient on e could be consistent with a buffer-stock model, depending on, among
other factors, the degree of correlation of the various instruments with the transitory and
permanent components of growth.
Whether in practice a buffer-stock model predicts something like the 0.5 coefficient that
- 25 -
Campbell and Mankiw typically found would depend in detail on the exact specification of the
model; setting up and solving such a model is well beyond the scope of this paper (although it is
an inviting project for future work). The argument here is only that a buffer-stock model at least
has the potential to explain Campbell and Mankiw’s results without assuming the existence of
consumers who blindly set consumption equal to income in every period.
One important caveat in using the model to explain aggregate data is that, even if the
typical household is a buffer-stock consumer, it is clear that at least some consumers do not
behave according to a buffer-stock model (see the discussion of very wealthy households in
section IVC). To the extent that aggregate consumption reflects the behavior of these nonbuffer-stock consumers, the buffer-stock model may not be able to match aggregate data even if
it is the right model for most consumers.
III. E. Other Implications
The results in Table I summarize other interesting characteristics of the steady-state
solution of the buffer-stock model under a variety of parameter values.
First, the Marginal Propensity to Consume.
The standard model under baseline
parameter values implies an MPC out of transitory income of 2 percent. No commonly used set
of parameter values in the standard model implies an MPC of greater than about 8 percent (the
formula for the MPC in the standard models is given in Section II.C.). Table I shows that, over
the entire range of parameter values considered in the table, the MPC is much greater than for the
standard model: The average MPC for buffer-stock consumers is always at least 15 percent, and
ranges up to 50 percent.
Another interesting question is what the model implies about the relationship between
expected income growth and the personal saving rate. If the steady-state average net wealth ratio
is w* at income growth rate g, then (if the interest rate is zero) the personal saving rate necessary
to make wealth grow at rate g (thus keeping the wealth/income ratio constant) is s ≈ g w*. If the
target wealth ratio w* were a constant, the model would obviously imply that s ≈ g w* is higher
when g is higher. However, w* is a negative function of g; the standard term for the effect of g
on consumption and saving decisions is the “human wealth effect.” In practice, for all parameter
values considered here the human wealth effect is vastly smaller than in the standard model, so
- 26 -
across steady-states the elasticity of the saving rate with respect to the growth rate of income is
positive.
In addition to its implications for the relationship between saving and growth, the drastic
diminution of the human wealth effect compared to the standard model is also an interesting
finding in itself. Several recent empirical tests can be interpreted as providing evidence that
current consumption is affected much less by expected future income than the standard LC/PIH
model implies. Campbell and Deaton [1989] calculate the implied effect on human wealth of an
innovation in current income, and find that consumption greatly ‘underresponds’ to innovations
in human wealth. Carroll [1994] projects mean future income for a panel of households and
finds no evidence that predictable future income growth affects current consumption at all.
Viard [1993] shows that a standard LC/PIH model implies that the post-1973 slowdown in
productivity growth in the U.S. should have sharply boosted saving rates; instead, saving rates
have fallen.
Is the MPC out of human wealth in a buffer-stock model consistent with these results?
Unfortunately, such a question is difficult to answer because, in contrast to the standard LC/PIH
model, in precautionary saving models the MPC out of future income depends on the current
level of physical wealth, the distribution of the future income, and the consumption rules
expected to prevail over the entire remainder of the consumer’s horizon. There is thus no
general way to answer the question of how responsive consumption is to human wealth, because
the answer will depend on the exact experiment. It is possible, however, to answer a specific
question, such as how consumption would respond to an increase in the expected growth rate of
income from 2 percent to 3 percent per year. Consider Figure III, which presents the converged
consumption rules for infinite-horizon versions of the buffer-stock model and the certainty
version of the LC/PIH model in the case where the real interest rate is four percent, expected
income growth is either two percent or three percent, and other parameters are at their baseline
values.23 The straight lines indicate the optimal consumption functions in the perfect certainty
case; the curved functions represent the converged buffer-stock consumption rules. The dashed
23
The deviation of the interest rate from the baseline value of zero is motivated by the fact that the present discounted value of
future income in the certainty case is infinite, making nonsense of the model.
- 27 -
functions correspond to the g=.02 cases and the solid functions to the g=.03 cases. When the
growth rate of income increases from two to three percent, human wealth defined as the present
discounted value of expected future labor income doubles,24 boosting consumption substantially
at all levels of gross wealth in the standard model.
consumption rises far less:
In the buffer-stock model, however,
Prudent buffer-stock consumers refuse to spend much out of
expected higher future income, because that income just might not materialize.
The same point is made numerically by Table II. Under the chosen parameter values the
MPC out of human wealth in the standard model is 3.8 percent; in the buffer-stock model the
MPC rises with current assets but remains very small over the entire range. Although not
precisely zero, it is easily small enough to be consistent with empirical estimates like those in
Carroll [1994] that indicate that the MPC out of human wealth is close to zero.
Another way to measure the degree of responsiveness of current consumption to changes
in expected future income is to calculate an implicit interest rate at which expected future income
can be said to be “discounted” when the consumer decides how much of that future income to
spend today. Appendix III describes how such a measure can be calculated; the results are
presented in the column entitled “Implied Discount Rate for Future Income” in Table II. The
results are striking: at a gross wealth ratio of 0.2, the implied rate at which future income is
discounted is 13,981 percent! This result arises because a consumer with gross wealth of only
0.2 is already consuming almost every penny, and when expected future income goes up, this
consumer is unable to spend more than a tiny bit extra today.
Even at more moderate values of the gross wealth ratio, the implied discount rate is
remarkably large. In particular, the target gross wealth ratio in this model is around 1.6; the
future income discount rate at a gross wealth ratio of 1.6 is about 22 percent.
IV.
Buffer-Stock Saving and the Life Cycle: Resolving Three Empirical Puzzles
This section of the paper makes a systematic argument that a version of the LC/PIH
model which implies that consumers engage in buffer-stock saving behavior over most of their
working lifetimes fits the essential facts about the behavior of the typical household’s
24
This can be seen from the formula for human wealth in the infinite-horizon case in Section II.C.: H = Y/(r-g). At r=.04, g=.02,
H = 50 Y; at g=.03, H = 100 Y.
- 28 -
consumption, income, and wealth better than either the “standard” LC/PIH model, a Keynesian
alternative, or the hybrid of the Keynesian and standard LC/PIH models proposed by Campbell
and Mankiw [1989].25
The buffer-stock version of finite-horizon LC/PIH model will reflect the same baseline
parameter values under which buffer-stock saving behavior emerges in the infinite-horizon
context, but with a lifetime income process calibrated to actual US household age/income
profiles. The perfect-certainty finite-horizon model briefly described in Section II.C. is what I
will call the “standard” LC/PIH model, although similar results would be obtained with a model
with quadratic or Constant Absolute Risk Aversion utility. The Keynesian model will be taken
to be a model of the form C =
0
+
1Y
+ u, where consumption has a positive intercept
there is a constant marginal propensity to consume
1
0
and
that is close to one (perhaps .9). The
Keynesian model has not been taken seriously since the work of Friedman and Modigliani in the
1950s; it is examined here principally as an aid to understanding the nature of the evidence. The
final model is the Campbell-Mankiw model which blends the standard LC/PIH and the
Keynesian models by assuming that half of income goes to standard LC/PIH consumers and half
goes to Keynesian consumers with
0
= 0 and
1
= 1.
Table III summarizes some of the principal points made below. I will argue that only a
buffer-stock version of the LC/PIH model is consistent with the overall pattern of facts.
IV.A.The Consumption/Income Parallel in Low Frequency Data
Perhaps the most striking point of Carroll and Summers [1991] is made by Figure IV:
Across occupations, differences in age/income profiles are closely paralleled by differences in
age consumption profiles.26,27 The figure shows that consumption growth and income growth are
25
I do not consider a model with uncertain income and liquidity constraints, as in Deaton [1991], because I view that model and
the model presented here as close substitutes. Most of the evidence that I find to be consistent with the buffer-stock model
presented here would also be consistent with Deaton’s model.
26
Carroll and Summers [1991] and Carroll [1994] provide evidence that the profiles for different occupational groups remain
relatively stable over time.
27
Unlike the figure in Carroll and Summers [1991], income and consumption in this figure were adjusted for aggregate
productivity growth by adding 1.5 percent to the growth of both the income and the consumption profiles in each year. Without
this adjustment for aggregate productivity growth, Carroll and Summers found that for all occupations income reached a peak in
middle age and then began declining, rapidly in some cases.
- 29 -
very closely linked over periods of a few years or longer, a phenomenon we dubbed the
“consumption/income parallel.” This phenomenon is interesting because there is no explanation
for it in the unconstrained standard LC/PIH framework; in that model the pattern of consumption
growth is determined by tastes and is independent of the timing of income.
Of course, the Keynesian model C =
consumption/income parallel if
0
is small and
1
0
+
1Y
+ u can easily explain the
is near one. Little of the evidence Carroll and
Summers marshalled for the low-frequency consumption/income parallel would rule out such an
explanation; however, evidence presented in the next section (along with a mountain of other
evidence originally presented in the 1940s and 1950s by Modigliani, Friedman and others) is
much less favorable to the Keynesian model.
The Campbell-Mankiw model with
= .5 has trouble explaining the consumption/income
parallel because the parallel is simply too close. The ocular regression of consumption on
income in Figure IV suggests a coefficient near 1, not near 0.5; the regressions of Carroll [1994]
also suggest coefficients much nearer 1 than 0.5.
Can a plausibly parameterized buffer-stock version of the LC/PIH model explain the low
frequency consumption/income parallel? To answer this question I simulate a finite-horizon
version of the model using income profiles calibrated to roughly match those in Figure IV. From
Figure IV it is appears that in some occupations, such as Unskilled Labor, labor income tends to
stop growing relatively early in life, while for others, such as Managers, income tends to
continue growing until late middle age; for other occupations, such as Operatives, income grows
quickly until middle age, then slowly until retirement. For the simulations, I will consider three
age/income profiles roughly calibrated using the data for Unskilled Laborers, Operatives, and
Managers shown in Figure IV. Specifically, the Unskilled Labor profile income grows at 3
percent annually from ages 25 to 40, and is flat from age 40 to retirement at 65. For Operatives,
labor income grows at 2.5 percent per year from age 25 to age 50, and then at 1 percent per year
until retirement. Finally, for Managers income grows at 3 percent a year from ages 25 to 55, and
declines at 1 percent a year from 55 to 65. Post-retirement income for all three categories is
The productivity adjustment does not alter the principal conclusion Carroll and Summers [1991] drew from their figure
(and from a variety of other evidence):
- 30 -
assumed to equal 70 percent of income in the last year of the working life, because empirical
estimates suggest that the sum of pension, social security, and other noncapital income after
retirement typically falls to roughly 70 percent of its preretirement level.
Solving the finite horizon version of the model by backwards recursion on (4) produces
optimal consumption rules for each period of life. Estimates of average consumption and
income were generated as for Figure II and Table I by randomly drawing income shocks
according to the assumed income distributions, for 1000 consumers who start life with zero
assets.28 (The behavior of asset holding over the life cycle is discussed below).
The results are shown in Figure V. A rough summary of the results of Figure V would be
that consumers engage in buffer-stock saving behavior, and so consumption growth closely
parallels income growth, until roughly age 45 or 50. Around that age consumers switch over to
doing a bit of retirement saving, which allows the income profile to rise a somewhat above the
consumption profile in the years immediately before retirement.
It worth noting here that Friedman himself would not necessarily have interpreted Figure
IV as evidence against his conception of the PIH. Indeed, he almost seems to anticipate it:
For any considerable group of consumers the ... transitory components tend to average
out, so that if they alone accounted for the discrepancies between permanent and measured
income, the mean measured income of the group would equal the mean permanent component,
and the mean transitory component would be zero. ...
... It is tempting to interpret the permanent components as corresponding to average
lifetime values and the transitory components as the difference between such lifetime values and
the measured values in a specific time period. It would, however, be a serious mistake to accept
such an interpretation... (pp. 22-23)
The permanent income component is not to be regarded as expected lifetime earnings; it
can itself be regarded as varying with age. It is to be interpreted as the mean income at any age
regarded as permanent by the consumer unit in question, which in turn depends on its horizon and
28
The slight upward fillip to consumption in the last two or three years of life occurs as consumers spend down their
precautionary assets when they realize that the amount of uncertainty remaining is small. This feature is an artifact of the
unattractive assumption of a certain date of death, and therefore is not one of the implications of the model on which I wish to
concentrate. If the model were modified to incorporate length-of-life uncertainty by adding a probability of death in each year, it
could likely replicate the results in Hubbard, Skinner, Zeldes [1995] who find that consumption slopes downward throughout the
entire retirement period.
- 31 -
foresightedness. 29 (p. 93)
- Milton Friedman, A Theory of the Consumption Function
Friedman explains that the principal reason interpreting “permanent income” as “average
lifetime income” is a mistake is because such an interpretation prejudges the issue of the length
of the “horizon” over which consumers calculate permanent income. Friedman later clarifies
what he means by the “horizon” when he says that the typical household discounts future income
at a rate of 33 1/3 percent. Although discounting future income at a 33 1/3 percent rate is very
difficult to justify in the standard model, as Table II shows, in a buffer-stock model an apparent
“discount” rate of 33 1/3 percent is not at all difficult to obtain.
IV.B. The Consumption/Income Divergence in High Frequency Data
It might seem that the low-frequency parallel between consumption and income suggests
that a simple Keynesian model with
0
near zero and
1
near one is a better model than the
comparatively complicated buffer-stock or standard LC/PIH models. The Keynesian model,
however, was abandoned in the 1950s for a number of good reasons, perhaps the most important
of which was the discrepancy between estimates of the model using long-term aggregate data,
which found
0
near zero and
1
near one, and estimates using cross-section household surveys
of consumption and income, which found
0
to be substantially positive and
1
to be much less
than one.
Friedman’s [1957] famous resolution of this puzzle was one of the persuasive pieces of
evidence for the Permanent Income Hypothesis. He showed that if C = P + u and Y = P + v,
where P represents permanent income and Y represents observed income and u and v are
stochastic, then a regression of C on Y would produce an estimated
1
coefficient on observed
income of far less than one even if the MPC out of permanent income is one. The logic is
identical to the now-familiar errors-in-variables econometric problem in which the coefficient on
a variable measured with error is biased toward zero. To complete the case, Friedman argued
29
Another indication that Friedman regards mean income as good indicator of permanent income is his statement that upon
retirement “permanent income” typically falls by about 25% because pension income is less than working income.
- 32 -
that the reason
1
was estimated to be much nearer one when long-term aggregate income data
were used was because such long-term data are dominated by changes in permanent income.
Friedman [1957] noted that another implication of the model was that groups within the
population for whom the variance of transitory shocks to income is relatively greater should have
a lower estimated marginal propensity to consume out of total income; as confirmation, he
presented evidence that farmers and entrepreneurs had a lower MPC than other households.
A closely related implication of the model is that groups of consuemrs with a greater
variance of transitory shocks should on average exhibit a greater divergence between
consumption and income. The simple Keynesian model with
0
close to zero and
1
near one
has no such implication. To see this, nest the two models by defining consumption as C = P +
v + u, where
is the MPC out of transitory income v. If = 0, this corresponds to the permanent
income model, and if
= 1 it corresponds to the Keynesian model with
0
= 0 and
1
= 1. Now
we have:
var(C/Y)
= var( (P + v+ u ) / (P + v) )
= var( (1 + v/P + u/P ) / (1 + v/P) )
≈ var( (1 + v/P + u/P ) (1 - v/P) )
= var( (1 + v/P + u/P - v/P -
(v/P)2 - (v/P) (u/P) )
≈ var( ( -1) (v/P) + (u/P) ),
where the first approximation holds when v/P is not too far from zero, and the second
approximation holds if (v/P)2 and (v/P)(u/P) are both close to zero, which will be true for
plausible estimates of the magnitude of transitory shocks to income and consumption.30
Assuming that the covariance between (v/P) and (u/P) is zero, this becomes
var(C/Y)
30
≈ var( ( -1) (v/P) ) + var(u/P) )
The standard deviation of logarithmic transitory shocks to income, which can be identified here with e/p, is estimated in Carroll
[1992] to be 10 percent annually; for a one standard-deviation shock, the error in the first approximation is given by comparing
1/1.1 ≈ .91 to .9. For the second approximation, if the standard deviation of consumption shocks is roughly the same size, both
(e/p)2 and (e/p)(u/p) should be small enough to safely ignore.
- 33 -
= ( - 1)2 var(v/P) + var(u/P).
(11)
Thus the Keynesian model ( = 1) implies that the variance of the consumption/income
ratio is unrelated to the variance of transitory shocks to income, while the PIH model ( = 0)
implies that the variance of the consumption/income ratio moves one-for-one with the variance
of transitory shocks to income.31
Table IV presents estimates of the variance of the consumption/income ratio by
occupation group calculated from the 1960-1961 Consumer Expenditure Survey, along with
estimates of the variance of transitory shocks to labor income taken from Carroll and Samwick
[1995a].32,33 It is clear that there is a strong positive association between the two variances across
occupation groups. A simple linear regression using the data in this table yields
var(C/Y)
= .651 var(v/P) - .0072
(.128)
(.0059)
which implies a value for , the MPC out of transitory income, of about 0.2. This estimate is
statistically significantly different from both zero and one at the five percent level of
significance.
These results constitute stronger evidence against the Keynesian model than against the
standard LC/PIH model, because the usual errors-in-variables logic shows that any measurement
error in the estimate of the variance of transitory income by occupation group would bias the
regression coefficient on the transitory variance toward zero, the value implied by the Keynesian
31
Although the simple Permanent Income model discussed here is not explicitly a finite-horizion model, similar results would
apply for the standard finite-horizon model described in section II.C.
32
The data in both datasets are for consumers between the ages of 25 and 50 -- that is, consumers in the age range where I argue
that the infinite-horizon version of the buffer-stock model is a good approximation to the behavior of finite-horizon consumers.
33
Carroll and Samwick [1995a] used essentially the same technique to decompose shocks to income into transitory and
permanent components that Carroll [1992] and Hall and Mishkin [1982] used. This technique is unable to distinguish transitory
shocks to income from white noise measurement error in income. However, the point of the table is not related to the level of the
transitory variance, but rather to the differences in estimated transitory income across groups. If the variance of measurement
error is the same across occupation groups, the coefficient on the regression (and the implied estimate of the MPC out of
transitory income) is unbiased.
- 34 -
model. The coefficient is very significantly different from zero despite this bias toward zero, so
the rejection of the Keynesian model is even stronger than implied by the simple t-statistic.
The same bias, however, weakens the case the regression makes against the standard
LC/PIH model; it is conceivable that errors-in-variables bias is the only reason the MPC out of
transitory income was estimated to be 0.2 rather than zero. Of course, this regression is not the
only evidence that the MPC out of transitory income is too large to be consistent with the
standard LC/PIH model. Early tests of the permanent income hypothesis found that the marginal
propensity to consume out of unexpected, nonrecurring windfall payments was somewhere
between .15 and .5. (see, e.g., Bodkin [1959], Kreinin [1961], and the discussion by Mayer
[1972]).34 Similar results have been found by many subsequent authors proceeding through Hall
and Mishkin [1982] to very recent work by Nicholas Souleles [1995].35
Table I showed that the average MPC out of transitory income is 15 percent or greater in
the buffer-stock model for all combinations of parameter values considered in the table. Similar
MPCs arise over most of the working lifetime in the finite-horizon version of the buffer-stock
model; results are not reported to conserve space. Of course, other parameter values could
generate either larger or smaller MPC’s; the essential point is that the buffer-stock model has no
difficulty generating an MPC large enough to match even the larger empirical estimates, while,
for plausible parameter values, the standard LC/PIH model is simply incapable of implying large
values for the MPC out of transitory income.
It is again worth noting that Friedman would not have found these results inconsistent
with his understanding of the Permanent Income Hypothesis. Friedman [1963] states that the
Permanent Income Hypothesis implies a marginal propensity to consume out of purely transitory
income of about 0.3.
IV.C. The Behavior of Wealth Over the Lifetime
34
Two natural experiments were examined in these papers: in the U.S., the response of consumption to the “National Service Life
Insurance Dividend of 1950,” a special payment to World War II veterans, and in Israel, reparations payments to victims of
German persecution during World War II. The authors argue that both were unanticipated and transitory shocks to income.
35
Although they interpret their results as suggesting that 20 percent of consumption is done by consumers who set consumption
equal to income, an alternative interpretation is that all consumers have a marginal propensity to consume out of transitory
income of 20 percent.
- 35 -
Diamond and Hausman [1984] called attention to the fact that the median household
typically holds surprisingly small, but still positive, amounts of financial wealth over the entire
working lifetime. Table V illustrates this point using data from the 1963, 1983, and 1989
Surveys of Consumer Finances. In all three surveys, at all ages before retirement, the ratio of
median financial assets to median annual income is between 2 percent and 35 percent. In all
three surveys the ratio rises modestly from ages 25 to 55, and then rises sharply in the last decade
before the typical retirement age of 65.
The fact that the median ratio of wealth to income stays within a rather narrow range until
just before retirement is not necessarily inconsistent with the standard LC/PIH model. Indeed,
any particular pattern of wealth accumulation over the lifetime can be justified by some set of
assumptions about tastes and the pattern of income over the lifetime. The stability of the
Diamond-Hausman phenomenon between the early 1960s and the late 1980s, however, is
troubling for the standard LC/PIH model, because at some point between these two surveys there
was a sharp slowdown in productivity growth and expected future productivity growth [Viard,
1993].
To explore the implications of the standard and buffer-stock versions of the LC/PIH
model for the age profile of the wealth ratio, I solved both models for the age-income profile
labelled “Operatives” in Figure V, which was calibrated using the 1960s data. Under baseline
parameter values, the standard LC/PIH model implies large negative wealth holding over most of
the life cycle; however, there are obviously other parametric assumptions under which the model
implies positive wealth.
I experimented with the assumed interest rate and found that an
assumed interest rate of eight percent per year produced the age/wealth profile that most closely
resembled the pattern in Table V.
The result is the age-wealth profile labelled “Faster
Productivity Growth” in Figure VI. I then solved the model again assuming that the post-1973
productivity growth slowdown resulted in a one percent slower growth rate of labor income over
the working lifetime.36
The resulting age/wealth profile is labelled “Slower Productivity
Growth.” The difference between the two curves is a rough measure of how the productivity
36
See Carroll and Summers [1991] for some evidence that this is a good approximation of reality.
- 36 -
slowdown should have affected the age/wealth profile for U.S. consumers if the standard LC/PIH
model were correct. The figure shows that lower growth should have induced an enormous
increase in household wealth at all ages greater than 30. On average, the “Slower” curve is
higher than the “Faster” curve by an amount equal to roughly two years’ worth of income.
Comparison to Table V indicates that the model’s predicted enormous increase in household
wealth/income ratios is more than an order of magnitude greater than the actual increase in
wealth/income ratios.
The results for the same experiment with the buffer-stock model are shown in Figure VII.
The profile of the wealth ratio over the lifetime in this figure bears a strong resemblance to the
profiles shown in Table V. Very early in life, the wealth ratio is low. For the middle two
decades it grows slightly, and then in the last decade before retirement the wealth ratio grows
sharply. The ratio is uniformly a bit higher in the economy with slow productivity growth, but
the difference between the two wealth profiles is not remotely so dramatic as in the standard
model. The implication of a slightly higher wealth ratio in the slow-growth equilibrium is
consistent with the pattern in Table V: for every age category, the wealth ratios are a bit higher
for the 1983 and 1989 surveys than for the 1963 survey.
Of course, the correspondence between the figure and the tables is not perfect. Under the
chosen parameter values, the buffer-stock model implies a considerably larger buffer-stock than
is evident in Table V. The model’s predictions could be brought more in line with the results in
Table V by assuming consumers are more impatient, or less risk averse, that they face less
uncertainty, or that they face a faster growth rate of income.
One objection to this line of argument might be that comparison to Table V is not
appropriate, because the standard LC/PIH model’s implications are about total net worth, not
about financial assets. However, similar calculations comparing the ratio of net worth to income
in the 1963, 1983, and 1989 survey produce similar conclusions: wealth-to-income ratios
increased only slightly from the 1960s to the 1980s, rather than by the enormous amounts the
standard model would imply.37
37
It would not be appropriate to include, say, expected future pension benefits in the measure of wealth. As shown in Table 2, in
this model there is no single interest rate at which it is appropriate to discount uncertain expected future income.
- 37 -
One feature of the model that appears to be strongly at variance with available evidence
is its implication that wealth falls sharply after retirement, reaching zero in the year of death.
This implication is a result of the assumptions that the date of death is known with certainty, that
there is no bequest motive, and that forms of uncertainty other than labor income uncertainty
(such as uncertain medical expenses) do not intervene to boost the saving rate as consumers
age.38 Because it does not address these issues, the buffer-stock model is probably less useful in
understanding the behavior of the elderly than it is for describing the working population,
particularly the working population well before retirement age.
Less formal evidence about the reasons for holding wealth may also be illuminating. As
noted in the introduction, many more people cite “emergencies” than “retirement” as the most
important reason for saving. Another informal source of evidence about how people think about
uncertainty and savings is provided by personal financial planning guides.
These guides
commonly have passages that suggest that consumers should maintain a buffer stock of assets
against uncertainty. The following is a typical passage:
It is generally held that your liquid assets should roughly equal four to six months'
employment income. If you are in an unstable employment situation ... the amount should
probably be greater.
The Touche Ross Personal Financial Management and Investment Workbook, 1989, p. 10.
As both this quotation and intuition suggest, one of the implications of a buffer-stock
model is that consumers with higher income uncertainty should hold more wealth. Several
recent papers have found empirical evidence that precautionary saving is statistically significant
and economically important. Using wealth and income uncertainty data from the PSID, Carroll
and Samwick [1995a,b] find that wealth is substantially higher for consumers who face greater
income uncertainty. Carroll [1994] provides some evidence that consumers with more variable
incomes save more.
38
Kazarosian [1990] finds, in a regression of wealth on demographic
The certain date of death accounts for the upward fillip in consumption in the last two or three years of life. As the certain end
of life approaches, uncertainty about future income approaches zero, so remaining precautionary assets are spent.
- 38 -
characteristics and income variability, that the degree of income variability is overwhelmingly
significant.
One further category of evidence on household wealth accumulation patterns supports the
buffer-stock interpretation of the LC/PIH model. This is observation by Avery and Kennickell
[1989] that wealth holdings are extremely volatile, even over short periods. They first attempt to
explain the changes in consumers' wealth between 1983 and 1986 with a statistical life cycle
model, but the model performs poorly, explaining at most about 8 percent of the observed
changes in wealth. The reason for the failure is that in the standard life cycle model wealth
changes glacially, gradually accumulating or decumulating depending on life cycle stage, while
in the SCF data wealth appears to fluctuate vigorously, just as would be expected in a model
where the primary purpose of holding wealth is as a buffer against random shocks to income.
Even if the buffer-stock model provides a good description of the patterns of financial
asset accumulation of the typical household, it is nevertheless clear that it is not a complete
model of household wealth accumulation. For the typical household, housing equity constitutes
a larger fraction of total net worth than do financial assets, and the buffer-stock model, with its
single perfectly liquid, perfectly riskless asset, is a poor vehicle for understanding housing
investments. One interpretation of the strong correspondence between the buffer-stock model’s
implications and observed patterns of financial asset holding and consumption behavior is that
consumers separate the housing decision from other consumption-saving decisions. Once they
have bought a house and committed themselves to a fixed monthly mortgage payment, they may
subject the remaining “disposable” income stream to buffer-stock saving rules.
Finally, even as a model of financial asset holding, the buffer-stock model cannot be
considered a complete description of the behavior of all households. This is illustrated by two
related facts. First, if all consumers behaved according to a buffer-stock model under the
baseline parameter values assumed here, the aggregate capital-income ratio would be far smaller
than we observe it to be. Second, the distribution of financial asset holdings is far more
concentrated than the model implies. For example, in 1983 the richest 1 percent of households in
the US held 64 percent of total financial assets held directly by the household sector. These
people are clearly not buffer-stock savers, but it seems unlikely that they are life cycle savers
- 39 -
either. To be complete, any description of the determinants of aggregate wealth must capture the
behavior of these consumers.
V. Literature Survey
In the last few years a substantial literature has appeared examining the implications of
models similar to the one in this paper. This section provides a brief discussion of some of the
recent literature.
Two papers by Hubbard, Skinner, and Zeldes [1994, 1995] (henceforth, HSZ) examine
the theoretical properties of a generalized version of the model in this paper. The principal
generalizations are that they incorporate health risk and mortality risk, and they carefully model
the US social insurance system. They calibrate medical expense risk and mortality risk using
empirical data, but find that neither health risk nor mortality risk has much effect on behavior,
given the presence of labor income risk. Their empirical estimates of labor income risk are
similar to those in this paper, and when my model is calibrated using the HSZ parametric
assumptions, the two models generate similar predictions about lifetime age/wealth and
age/consumption profiles.
The most important difference in parametric assumptions is that HSZ assume that
consumers face substantially slower income growth. This is largely because their age/income
profiles are estimated in a way that removes any aggregate productivity growth from their
estimated household income process.
Carroll and Summers [1991] provide a variety of
evidence, however, that in the medium- and long-run, household income shares in aggregate
productivity growth.
One advantage of the HSZ parameterization is that it causes the model to generate
substantially larger estimates of aggregate wealth than it generates under my parameterization
(henceforth, the buffer-stock parameterization). However, the implied distribution of wealth
across households under HSZ parameter values differs greatly from the actual empirical
distribution. In particular, the model under the HSZ parameterization substantially overpredicts
the wealth of the median household over most of the lifetime, but greatly underpredicts the
wealth of the richest households.
While the buffer-stock parameterization also greatly
underpredicts the wealth of the richest households, it appears to match median age-wealth
- 40 -
profiles better than the HSZ parameterization. A related point is that the HSZ parameterization
generates considerably less tracking of consumption to income over the lifetime than does the
buffer-stock parameterization.
A very recent paper by Gourinchas and Parker [1995] uses synthetic cohort data from the
U.S. Consumer Expenditure Surveys in a way that lets the available consumption and income
data determine the period over which consumers engage in buffer-stock saving behavior. They
find that consumers typically make the transition between buffer-stock saving and life-cycle
saving somewhere around age 45, at the low end of the range originally proposed in this paper.
They also use a model essentially identical to the one in this paper to estimate time preference
rate and risk aversion parameters by occupation and education group. They find substantial, and
intuitive, differentials in time preference rates across groups.
Other evidence which supports the buffer-stock parameterization is provided in Carroll
and Samwick [1995a]. We calculate the predictions of the model for the relationship between
income uncertainty and wealth holding under the buffer-stock parameter values and the HSZ
parameter values, and find that under the HSZ parameter values the model implies that wealth is
roughly an order of magnitude more responsive to uncertainty in permanent income than under
the buffer-stock parameter values. We then estimate the empirical relationship between saving
and uncertainty, and find coefficients similar to those implied by the buffer-stock
parameterization, and highly statistically different from those implied by the HSZ
parameterization. Another recent paper by Carroll and Samwick [1995b] uses the buffer-stock
model in combination with empirical methods to estimate that between a third and one half of the
wealth of the typical working household that is attributable to buffer-stock saving behavior.
Several other recent papers develop further implications of the infinite-horizon version of
the model. Heaton and Lucas [1994] examine implications of the model for portfolio choice.
Ludvigson [1996] develops a buffer-stock model with liquidity constraints which vary with the
consumer’s level of income, and uses the extended model to analyze the relationship between
household balance sheet positions and aggregate consumption growth in the U.S. Bird and
Hagstrom [1996] use data from the Survey of Income and Program Participation (SIPP) to test
the model's implication that more generous social insurance benefits reduce the target wealth
- 41 -
stock. Gross [1995] adapts the model to study the investment decisions of liquidty constrained
firms.
VI.
Conclusion
The standard version of LC/PIH model remains the most commonly used framework for
both micro and macro analysis of consumption behavior despite a large and growing body of
evidence that it does a poor job explaining those data. This paper argues that a version of the
LC/PIH model in which buffer-stock saving emerges is closer both to the behavior of the typical
household and to Friedman's original conception of the Permanent Income Hypothesis model.
The buffer-stock version of the model can explain why consumption tracks income closely when
aggregated by groups or in whole economies, but is often sharply different from income at the
level of individual households. Without imposing liquidity constraints, the model is consistent
with recurring estimates of a much higher MPC out of transitory income than is implied by the
standard LC/PIH model.
And it provides an explanation for why median household
wealth/income ratios are persistently small and have remained roughly stable despite a sharp
slowdown in expected income growth. Insights from analysis of the buffer-stock version of the
model may also help to explain many of the failures and anomalies of Euler equation models
estimated using both household and aggregate data.
The buffer-stock model does not, of course, explain all behavior of all consumers. The
Surveys of Consumer Finances show that a small number of wealthy consumers holds enormous
financial assets. A buffer-stock model is clearly not a plausible description of the behavior of
these people. Many other consmers explicitly engage in some form of life cycle saving behavior,
particularly in the form of participation in pension plans. The model may also not be useful for
understanding housing investments. Probably the appropriate place for the buffer-stock model is
as an explanation of truly discretionary “high frequency” saving decisions of the median
consumer. It seems plausible that many consumers ensure that retirement is taken care of by
joining a pension plan, buy a house, and then subject the post-pension-plan, post-mortgagepayment income and consumption streams to buffer-stock saving rules.
The
buffer-stock
version of the LC/PIH model provides a new way of looking at both microeconomic and
- 42 -
macroeconomic data on saving and consumption, and has many testable implications that differ
from those of the standard LC/PIH model that has dominated empirical and theoretical work
until recently. It promises to provide a fruitful framework for future work.
- 43 -
Appendix I
Details of the Methods of Solution
The dynamic stochastic optimization problem solved in this paper is characterized by a
fundamental equation (equation (4) in the text) of the form :
1
= R Et [ { ct+1[R [xt - ct] / G Nt+1 + Vt+1] G Nt+1 / ct } -
]
or, returning to marginal utility notation and multiplying both sides by u'(ct),
(I.1)
u'( ct )
=R
Et u' ( ct+1(R [ xt - ct] / G Nt+1 + Vt+1) G Nt+1 ).
As in Deaton [1991], the method of solution for the finite-horizon version of the model is
to recursively solve backwards from the last period of life, in which the optimal plan is to
consume all assets, cT(xT) = xT. Given a value of xT-1, and a method for computing the
expectation (see the next paragraph) on the right hand side of (I.1), numerical algorithms can
locate the cT-1 which satisfies (I.1). In period T-1, equation (I.1) was solved numerically for
the optimal value of consumption for a grid of m values for x, values xi. The numerical
approximation to the optimal consumption rule cT-1(xT-1) was then constructed by cubic
interpolation (after experimenting with both linear and quadratic interpolation) between the
values of the function at the m grid points. Given cT-1(xT-1), a grid of x values for period T-2
was chosen, the numerical solution at each xi was computed from equation (I.1), and cT-2[x]
was given by cubic interpolation, and so on.
As described in the text, I assumed that, with some probability p, income would be zero
in period t+1, Vt+1 = 0. If income is not zero, then Vt+1 and Nt+1 are distributed lognormally
with expected values (1+p) and 1, respectively.
In solving the model, the lognormal
distributions were truncated at three standard deviations from the mean, yielding minimum
- 44 -
and maximum values V, N, V, and N. Full numerical integration is extremely slow, so the
lognormal distributions were approximated by a ten-point discrete probability distribution.
The distance (V - V) was divided into 10 equal regions of size (V - V)/10 with boundaries
denoted Bj.
Associated with each of these regions was the average value of V within the
Bj+1
region, computed by calculating the numerical integral Vj =
Bj
V dF(V). The probability of
drawing a shock of value Vj is given by F(Bj+1) - F(Bj). An analogous procedure was used to
approximate the distribution of permanent shocks. This method is similar to the methods
used by Deaton [1991], Hubbard, Skinner, and Zeldes [1995], and others working in this
literature.
In solving the infinite horizon problem a convergence criterion is needed to determine
when successive c(x) functions are sufficiently close that the consumption rules may be said
to have converged. The criterion used was:
1
m
| ct(xi) - ct+1(xi) | < .0005,
where i indexes the elements of the grid of m x's. I found that for most purposes, a grid of
twelve to fifteen values for the x’s produced results indistinguishable from much finer grids,
so most simulations reported in the paper were calculated using a twelve point grid.
Similarly, I found that the results do not change perceptibly even if the convergence criterion
is tightened substantially relative to the .0005 criterion used for most reported results.
- 45 -
Appendix II
The Numerical Method for Calculating the Ergodic Distribution of Net Worth
Standard methods for proving ergodicity do not appear to be applicable to the
distribution of net worth in this problem, primarily because the state space is, in principle,
unbounded from above. This problem can be solved by mapping the infinite range of x into a
finite interval z using the mapping z = x/(1+x). It is then possible, using the numerical
consumption rule and the assumptions on the distributions of shocks, to construct a
discretized approximation to the transition matrix for z. The question of whether there is an
ergodic distribution for z can then be answered numerically: if one of the eigenvalues of the
transition matrix is 1, then an ergodic distribution for z exists, and will be given by the
eigenvector associated with the eigenvalue of one.39 With the ergodic distribution for z in
hand, it is a simple matter to “unmap” it to obtain the corresponding steady-state distribution
for x.
39
I am grateful to Angus Deaton for suggesting this method of finding the steady-state numerically.
- 46 -
Appendix III
The Implied Discount Rate for Future Income
This appendix describes how the implied discount rate for future income presented in
Table 2 is calculated. In the infinite-horizon, perfect certainty version of the CRRA LC/PIH
model, if income P is expected to grow according to Pt+1 = G Pt forever, and G < R, consumption
in period t is given by:
ct = [1 - R-1 (R )1/ ](Wt + Ht)
where (if G < R) human wealth Ht = Pt / (1 - G/R) and where all other variables are as defined in
the description of the basic model. Consider changing the expected growth of income from G1 =
1 + g1 to G2 = 1 + g2. The change in consumption that results from this change in expected
income growth is given by:
(III.1)
c = [1 - R-1 (R )1/ ] [Pt / (1 - G2/R) - Pt / (1 - G1/R)].
For a given value of
c, for a particular set of taste parameter values, and for particular
values of g1 and g2, one can search for the value of R, and therefore r = R - 1, which solves this
equation. If such a value exists, it will correspond to the interest rate at which at which a
consumer with certain future income would have to discount future income and consumption in
order to justify the chosen c. Table 2 takes the values of
c calculated from the buffer-stock
model at various levels of the gross wealth ratio and presents the value of r that solves equation
(III.1).
THE JOHNS HOPKINS UNIVERSITY
- 47 -
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